Question

Show that the equation ax + by + d = 0 represents a plane parallel to the z - axis. Hence, find the equation of a plane which is parallel to the z - axis and passes through the points A(2, - 3, 1) and B(- 4, 7, 6).

Answer 1

Given – The equation of the plane is given by ax + by + d = 0 

To prove – The plane is parallel to z - axis 

Tip – If ax + by + cz + d is the equation of the plane then its angle with the z - axis will be given by sin^-1 \((\frac{c}{\sqrt{a^2+b^2+c^2}})\)

Considering the equation, the direction ratios of its normal is given by (a, b, 0) 

The angle the plane makes with the z - axis = sin^-1[0/√(a² + b²)] = 0

Hence, the plane is parallel to the z - axis 

To find – Equation of the plane parallel to z - axis and passing through points A = (2, - 3, 1) and B = (- 4, 7, 6) 

The given equation ax + by + d = 0 passes through (2, - 3, 1) & (- 4, 7, 6)

Substituting the values of a and b in eqn (i), we get, 

- 2Х10α + 3Х6α + d = 0 i.e. d = - 2α 

Putting the value of a, b and d in the equation ax + by + d = 0, 

(- 10α)x + (- 6α)y + (- 2α) = 0 

i.e. 5x + 3y + 1 = 0